Application of Derivatives

Quote:

Originally Posted by pankaj

Let $f:R\to R$ be a twice differentiable function and suppose that for all $x\in R$ , $f$ satisfies the following two conditions:

(i) $|f(x)|\leq 1$

(ii) $|f''(x)|\leq 1$

Prove that $|f'(x)|\leq 2$

using Taylor's theorem, we know that for any $x \in \mathbb{R}$ there exists some $c \in \mathbb{R}$ such that $f(x+2)=f(x)+2f'(x)+2f''(c).$ therefore:

$2|f'(x)|=|f(x+2)-f(x)-2f''(c)| \leq |f(x+2)|+|f(x)|+2|f''(c)| \leq 4,$ and hence $|f'(x)| \leq 2.$

a similar argument gives us this: if $|f(x)| \leq a$ and $|f''(x)| \leq b,$ for all $x \in \mathbb{R},$ then $|f'(x)| \leq 2 \sqrt{ab},$ for all $x \in \mathbb{R}.$